Scudamore believes there is a will among the sport's main governing bodies to make it happen, but feels the Football Association may have to look to its own flagship cup competition for the solution.
"Unless someone is prepared to give something up, it is pretty hard," he told the Daily Telegraph.
"We are not inclined to reduce the number of clubs in the Premier League - if you were running a theatre and had 380 nights that you wanted to sell, why would you throw 60 or 70 of those nights away?
"As for the FA, they don't want to give up replays in the FA Cup, so we all sit down and we all look at each other."
England manager Roy Hodgson has followed previous national team bosses in calling for a mid-season break, with the aim of ensuring players do not go into major tournaments having played for 10 months solid.
Scudamore added: "Nobody has asked the Football League to give up the League Cup, but why would we? We are huge supporters of it and we wouldn't have the temerity to alter the FA Cup either.
"That will be up to the FA, but unless the FA decide they have different priorities, where the England team is more important than something else they own, then that might be for them to look at."
- the Football Association